
where \(w_c=\dim W_c\) .
Only when \(\mathsf {Out}(G)\)  is an abelian group does this defect come from a
symmetry, as in that case (REF )
decomposes into a direct sum of invertible defects. For non-abelian
groups \(\mathsf {Out}(G)\)  we need the generalised orbifold construction
of [1]}, [2]} to go backwards.
The reason why the choice of defect (REF ) works is that the character of the regular representation \({L}^2(\mathsf {Out}(G))\)  of \(\mathsf {Out}(G)\)  is given by
\(\chi _{{L}^2(\mathsf {Out}(G))}(\kappa )= |\mathsf {Out}(G)| \, \delta _{\kappa ,\text{id}_G}\) 
for \(\kappa \in \mathsf {Out}(G)\) . Inserting a Wilson
loop corresponding to \(\mathcal {A}_G\)  into the path integral for a Riemann surface \(\Sigma \)  localises the integration domain to \(G\rtimes \mathsf {Out}(G)\) -bundles with trivial \(\mathsf {Out}(G)\)  holonomy around the inserted loop.
If at least one Wilson loop for every generator of the fundamental
group \(\pi _1(\Sigma )\)  labeled by \(\mathcal {A}_G\)  is inserted into the path integral,
then the \(\mathsf {Out}(G)\)  part of all bundles contributing to the partition function
is trivial and hence the partition function reduces to the partition
function (REF ) of Yang-Mills theory
with gauge group \(G\) . We will prove this rigorously in Section 
using the orbifold completion of the topological defect bicategory of
two-dimensional Yang-Mills theories.
